An optimal insulation problem

TL;DR

This paper investigates an optimal insulation problem where the goal is to find the best geometry of an insulator around a conductor to minimize heat dispersion, revealing that certain shapes like disks and balls are worst-case configurations under specific constraints.

Contribution

The paper introduces a new variational model for thermal insulation optimization and identifies geometric shapes that maximize heat retention under given constraints.

Findings

Disks are the worst shapes under perimeter constraints in 2D.

Balls are the worst shapes in higher dimensions under certain constraints.

The analysis provides insights into optimal insulation design and open problems for future research.

Abstract

In this paper we consider a minimization problem which arises from thermal insulation. A compact connected set $K$, which represents a conductor of constant temperature, say $1$, is thermally insulated by surrounding it with a layer of thermal insulator, the open set $\Omega\setminus K$ with $K\subset\bar\Omega$. The heat dispersion is then obtained as \[ \inf\left\{ \int_{\Omega}|\nabla \varphi|^{2}dx +\beta\int_{\partial^{*}\Omega}\varphi^{2}d\mathcal H^{n-1} ,\;\varphi\in H^{1}(\mathbb R^{n}), \, \varphi\ge 1\text{ in } K\right\}, \] for some positive constant $\beta$. We mostly restrict our analysis to the case of an insulating layer of constant thickness. We let the set $K$ vary, under prescribed geometrical constraints, and we look for the best (or worst) geometry in terms of heat dispersion. We show that under perimeter constraint the disk in two dimensions is the worst one. The same is true for the ball in higher dimension but under different constraints. We finally discuss few open problems.